Problem 9 - Advanced Engineering Mathematics

Question

Use Green's theorem to evaluate the given line integral. $$ \begin{aligned} &\oint_{C} 2 x y d x+3 x y^{2} d y \text { , where } C \text { is the triangle with vertices }\\\ &(1,2),(2,2),(2,4) \end{aligned} $$

Step-by-Step Solution

Answer

Integrate the double integral over the triangle's region to find the result using Green's Theorem.

Step 1: Identify Components

For Green's theorem, identify \( P(x, y) \) and \( Q(x, y) \) from the line integral \( \oint_{C} 2xy \, dx + 3xy^2 \, dy \). Here, \( P(x, y) = 2xy \) and \( Q(x, y) = 3xy^2 \).

Step 2: Find Partial Derivatives

Compute the partial derivatives needed for Green's theorem: \( \frac{\partial Q}{\partial x} = 3y^2 \) and \( \frac{\partial P}{\partial y} = 2x \).

Step 3: Set Up Green's Theorem

Apply Green’s Theorem: \( \oint_{C} P \, dx + Q \, dy = \iint_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \), where \( R \) is the region inside \( C \).

Step 4: Express and Evaluate the Difference

Substitute the partial derivatives into the formula: \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3y^2 - 2x \).

Step 5: Determine the Region’s Limits

The triangle has vertices \((1,2), (2,2), (2,4)\), which implies horizontal bounds are from \(x = 1\) to \(x = 2\). For \(y\), it ranges from \(y = 2\) to the line connecting points \((2,2)\) and \((2,4)\), which is simply \(y=y\) (or \((x-1) = 1\)).

Step 6: Solve the Double Integral

Evaluate the double integral: \[ \int_{1}^{2} \int_{2}^{x+2} (3y^2 - 2x) \, dy \, dx. \]

Step 7: Integrate with Respect to \( y \)

First, integrate with respect to \( y \): \( \int_{2}^{x+2} (3y^2 - 2x) \, dy \). Compute: \[ \left[y^3 - 2xy \right]_{y=2}^{y=x+2} = \left((x+2)^3 - 2x(x+2) \right) - \left(2^3 - 2x \times 2\right). \]

Step 8: Simplify and Integrate with Respect to \( x \)

Simplify the expression from Step 7: \( ((x+2)^3 - 2x(x+2)) - (8 - 4x) \). Integrate with respect to \( x \): \[ \int_{1}^{2} ((x+2)^3 - 2x(x+2) + 4x - 8) \, dx. \]

Step 9: Compute the Integral Results

Carry out the integration over \( x \) to find the numerical value. This completes the use of Green's theorem.

Key Concepts

Understanding Line Integrals

A line integral allows us to add up a function's values along a curve or path in a vector field. Imagine tracing out a curve on a map and summing the values of a function (like height) at each point along the path. In our exercise, the line integral is expressed as \( \oint_{C} 2xy \, dx + 3xy^2 \, dy \). The curve \( C \) is a triangle with vertices at \((1,2), (2,2), (2,4)\). Here, we are adding up the values of the functions \( 2xy \) and \( 3xy^2 \) along the edges of this triangle.

Line integrals are crucial in physics and engineering because they can represent work done by a force field along a path, among other things. To handle these integrals, it often involves components such as \( dx \) and \( dy \), which refer to infinitesimal changes along the x and y directions.

The Role of Partial Derivatives

Partial derivatives help us understand how a function changes when we vary one variable while keeping others constant. In the context of Green's theorem, they allow us to switch from a line integral around a closed curve to a double integral over the area it encloses. In our example, we need the partial derivatives \( \frac{\partial Q}{\partial x} \) and \( \frac{\partial P}{\partial y} \).

Given \( Q(x, y) = 3xy^2 \), we compute \( \frac{\partial Q}{\partial x} = 3y^2 \). This derivative shows how the function \( Q \) changes with x while y is held steady. Similarly, for \( P(x, y) = 2xy \), \( \frac{\partial P}{\partial y} = 2x \). Partial derivatives are not just numbers; they are themselves functions that depend on the variables.

These derivatives become crucial when setting up Green's theorem, letting us shift from a difficult line integral to an often simpler double integral.

Double Integration Explained

Double integration lets us compute the accumulation of a function's values over a two-dimensional area. It's similar to stacking many thin sheets over an area and finding the total, akin to computing volume. In our scenario, the function \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \) needs to be integrated over the triangular region.

Here's how it works: you first integrate with respect to one variable (holding the other constant) and then with the second variable. For this exercise, the double integral \[ \int_{1}^{2} \int_{2}^{x+2} (3y^2 - 2x) \, dy \, dx \] covers our region between x and y limits defined by the triangle's vertices.

By evaluating the inner integral (first regarding y) followed by the outer integral (regarding x), we track how the value of \( 3y^2 - 2x \) accumulates across the given area.

Exploring Vector Fields

Vector fields assign a vector to every point in a plane or space, describing how a quantity varies over an area. Imagine walking through a forest with a temperature map, each point on the map showing a temperature vector. Here, the vector field is represented by the functions \( P(x, y) = 2xy \) and \( Q(x, y) = 3xy^2 \), which describe how the field behaves across the triangle.

Understanding these fields is crucial when using Green's theorem, as it relates the line integral around the boundary of a region to the double integral over the region itself. By studying the behavior of these fields, we're able to apply the theorem to convert a complex line integral into a more approachable double integral.

• Vector fields model real-world phenomena, like electromagnetic forces or fluid flow.
• Each vector in the field indicates a direction and magnitude of some quantity.

Using vector fields and Green's theorem together provides powerful tools for solving problems in physics and engineering.